If `z_(1),z_(2),z_(3)` are three points lying on the circle `|z|=2`, then minimum value of `|z_(1)+z_(2)|^(2)+|z_(2)+z_(3)|^(2)+|z_(3)+z_(1)|^(2)=`

Options

A. 6
B. 12
C. 15
D. 24

Explanation

To solve the problem, we need to find the minimum value of the expression:

\[ |z_1 + z_2|^2 + |z_2 + z_3|^2 + |z_3 + z_1|^2 \]

where \( z_1, z_2, z_3 \) are points on the circle defined by \( |z| = 2 \).

### Step 1: Express the terms in the equation

Using the property of modulus, we can express each term as follows:

\[ |z_1 + z_2|^2 = (z_1 + z_2)(\overline{z_1 + z_2}) = |z_1|^2 + |z_2|^2 + z_1 \overline{z_2} + \overline{z_1} z_2 \]

Since \( |z_1| = |z_2| = |z_3| = 2 \), we have:

\[ |z_1|^2 = 4, \quad |z_2|^2 = 4, \quad |z_3|^2 = 4 \]

Thus, we can rewrite:

\[ |z_1 + z_2|^2 = 4 + 4 + z_1 \overline{z_2} + \overline{z_1} z_2 = 8 + z_1 \overline{z_2} + \overline{z_1} z_2 \]

### Step 2: Write the full expression

Now substituting back into the original expression:

\[ |z_1 + z_2|^2 + |z_2 + z_3|^2 + |z_3 + z_1|^2 = (8 + z_1 \overline{z_2} + \overline{z_1} z_2) + (8 + z_2 \overline{z_3} + \overline{z_2} z_3) + (8 + z_3 \overline{z_1} + \overline{z_3} z_1) \]

This simplifies to:

\[ 24 + (z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) + (\overline{z_1} z_2 + \overline{z_2} z_3 + \overline{z_3} z_1) \]

### Step 3: Analyze the cross terms

Let’s denote:

\[ S = z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1} + \overline{z_1} z_2 + \overline{z_2} z_3 + \overline{z_3} z_1 \]

### Step 4: Use the Cauchy-Schwarz inequality

To minimize \( S \), we can apply the Cauchy-Schwarz inequality. Since \( |z_1| = |z_2| = |z_3| = 2 \), we can find that:

\[ |z_1 + z_2 + z_3|^2 \leq (|z_1| + |z_2| + |z_3|)^2 = (2 + 2 + 2)^2 = 36 \]

Thus, we have:

\[ |z_1 + z_2 + z_3|^2 = |z_1|^2 + |z_2|^2 + |z_3|^2 + 2S = 12 + 2S \]

### Step 5: Solve for S

From \( |z_1 + z_2 + z_3|^2 \leq 36 \):

\[ 12 + 2S \leq 36 \implies 2S \leq 24 \implies S \leq 12 \]

### Step 6: Find the minimum value

Substituting back into our expression for the minimum value:

\[ |z_1 + z_2|^2 + |z_2 + z_3|^2 + |z_3 + z_1|^2 \geq 24 + 2S \geq 24 + 2 \cdot (-12) = 24 - 24 = 12 \]

### Conclusion

The minimum value of the expression is:

\[ \boxed{12} \]